Below find a list of the topics to be discussed in class. The references refer to the lecture notes to be handed out. During the semester the order of presentation may change, the changes should show on this page.

Already treated in class:

• 25.10.99
  Introduction
  • Introduction, general information
  • Derivation of the heat equation, §4.1
  • General second order linear partial differential equations, §5.2, Table 5.1
• 1.11.99
  Using the Matlab PDE Toolbox
  • Introduction to Matlab, the demos and §7.1
  • Solve a heat conduction problem, §7.2
  • Fluid flow problem as homework assignment
• From minimisation to differential equations
  • 8.11.99
    Electrostatic problem as a minimisation problem §5.1
  • 15.11.99
    General quadratic functional, §5.2, §5.3
  • Minimal surface problem, §5.4
  • Extrema of functions, §1
• A FEM program, the algorithm and coding in Mathamatica
  • 22.11.99
    Find formulas for contributions to element stiffness matrix, §6.1.1 - §6.1.3
  • 29.11.99
    Use EasyMesh to generate a mesh, coding in Mathematica, §6.2
  • 6.12.99
    Combine the element stiffness matrices to obtain the global stiffness matrix, §6.1.4
    Take boundary conditions into account, §6.1.5
    Coding in Mathematica, §6.2
• Calculus of Variations in one Variable
  • 13.12.99
    Shortest connection between two points, §3.1.2
    
  • 20.12.99
    Fundamental Lemma, §3.1.1
    Functionals with f(x,u(x)) , §3.1.3
    
  • 3.1.00
    Euler Lagrange Equation, §3.1.4
    First integrals, §3.1.6
    Surface of revolution
    Quadratic functionals, §3.1.5
    
  • 10.1.00
    Hamilton's principle of least action, double pendulum, moving car with pendulum, §3.3
• FEM in one variable
  • 17.1.00
    FEM solution of one dimensional problem §4.4
  • 24.1.00
    Step by step solution of simple problem by FEM, §2.2
  • 31.1.00
    List of approximation errors for FEM solution
    Gauss integration on intervals, §4.5.2
    
  • 7.2.00
    Linear and quadratic interpolation, §4.5.1
    Construction of second order element, §4.5.3

Soon to be treated:

• Introduction
  • Introduction, general information
  • Derivation of the heat equation, §4.1
  • General second order linear partial differential equations, §5.2, Table 5.1
• Using the Matlab PDE Toolbox
  • Introduction to Matlab, the demos and §7.1
  • Solve a heat conduction problem, §7.2
  • Solving a three dimensional problem, using polar coordinates, §7.3
• From minimisation to differential equations
  • Electrostatic problem as a minimisation problem §5.1
  • Simplified version of a quadratic functional, §5.3
  • Minimal surface problem, §5.4
  • General minimisation problem, §5.2
  • Extrema of functions, §1
• Finite Elements in two Variables
  • General description of the algorithms, §6.1
  • Coding in Mathematica with the help of EasyMesh, §6.2
• Calculus of Variations in one Variable
  • Shortest connection between two points, §3.1.2
  • Fundamental Lemma, §3.1.1
  • Euler Lagrange Equation, §3.1.3, §3.1.4
  • Quadratic functionals, §3.1.5, connection to §5.3
  • Radial symmetric heat problem, §4.1.4 and §7.3
• Calculus of Variations in two Variables
  • Euler Lagrange Equation, §5.2
  • Quadratic functionals and linear differential equations §5.3
• Applications of Calculus of Variations (tentative)
  • First integrals, §3.1.6
  • Deflection of a string, §3.3
  • Hamilton's Principle with examples, §3.3
  • Deflection of a laser beam by a heat source, §3.5
• Finite Elements in one variable
  • Horizontal bar problem, §2
  • General quadratic functionals, §4.3
  • First order elements, §4.4
  • Second order elements, Gauss integration, §4.5
  • Code in Mathematica, §4.6
  • Examples, §4.7
  • Convergence issues, §4.8
• Further topics

Homework assignments

1. Coding
   • mandatory to pass the class
     • Write code comparable to §6.2. Use your favourite programming language, e.g. Octave, Matlab, Scilab, C, Pascal, ...
       The code has to produce correct results and has to be well documented.
   • facultative
     • Code to visualise results in Octave, Matlab, Scilab, C, Pascal, ...
     • Use Triangle instead of EasyMesh to generate the grid
     • Find an algorithm to replace the Matlab code tri2grid, including documentation
     • If code is written in Octave, then test different solvers for the system of linear equations, LU, Cholesky, SuperLU. The interface to SuperLU is already programmed, see SuperLU for Octave
     • If code is written in C or Octave then test different solvers for the system of linear equations, LU, Cholesky, SuperLU (a sparse solver)
2. Sample problems
   • mandatory to pass the class
     • s simple fluid flow problem, to be solve with the Matlab toolbox
     • specified during the semester and posted here
   • facultative
     • specified during the semester and posted here

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October 4, 1999 by Andreas.Stahel@hta-bi.bfh.ch
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